1544
J. Chem. Theory Comput. 2006, 2, 1544-1550
Elimination of Translational and Rotational Motions in Nuclear Orbital Plus Molecular Orbital Theory: Contribution of the First-Order Rovibration Coupling Kaito Miyamoto, Minoru Hoshino, and Hiromi Nakai* Department of Chemistry, School of Science and Engineering, Waseda UniVersity, Tokyo 169-8555, Japan Received June 20, 2006
Abstract: The translation- and rotation-free nuclear orbital plus molecular orbital (TRF-NOMO) theory was developed to determine nonadiabatic nuclear and electronic wave functions. This study implemented a computational program for the TRF-NOMO method including first-order rotational terms, which corresponds to rovibronic coupling. Numerical assessments of first-order TRF-NOMO Hartree-Fock as well as second-order Møller-Plesset perturbation methods were carried out for several small molecules. The first-order contributions give small corrections in energy. Thus, we confirm that the approximate zeroth-order treatment is sufficient for eliminating the rotational contamination.
1. Introduction Born-Oppenheimer (BO) approximation,1 which divides nuclear and electronic motions, is a fundamental concept of modern molecular theory. Although BO treatment is suitable for accurately describing various chemical and physical phenomena, it cannot take into account coupling between the nuclear and electronic motions, that is, the non-BO effect. Adamowicz and co-workers2-5 proposed the non-BO theory by utilizing the explicit correlated Gaussian (ECG) basis functions, which involve the internal coordinates among nuclei and electrons. The ECG approach has been shown to achieve considerably high accuracy, that is, spectroscopic accuracy. The problem with this approach is that the complexity of the explicit formula increases with an increase of the number (N) of identical particles, which requires different programming codes for different N values. Furthermore, the computational cost increases very rapidly (N factorial). On the other hand, we have developed the nuclear orbital plus molecular orbital (NOMO) theory,6-12 which determines nuclear and electronic wave functions simultaneously without BO approximation. In NOMO theory, total wave function is constructed by nuclear orbitals (NOs), one-nucleus wave functions, and molecular orbitals (MOs), one-electron ones. We have proposed that it is convenient to adopt Gaussian basis functions with the center on each atomic position in * Corresponding author fax: +81-3-3205-2504; e-mail: nakai@ waseda.jp.
10.1021/ct6002065 CCC: $33.50
the appropriate molecular configuration, such as equilibrium and optimized ones.6-12 The use of Gaussian basis functions has been accepted by other groups13-28 probably because it is consistent with conventional MO theory within BO approximation. While Gaussian basis functions can describe a vibrational state accurately, translational and rotational states are not adequately reproduced because their motions are limited in some local regions represented by the functions. Thus, we have presented translation-free (TF)- and translation- and rotation-free (TRF)-NOMO theories and clarified the importance of eliminating translational and rotational contaminations in obtaining accurate results in NOMO and similar approaches.7,10-12 Sutcliffe has pointed out that it is possible to rigorously construct a TF-NOMO Hamiltonian, but it is impossible to rigorously construct a TRF-NOMO Hamiltonian for general systems.29 This distinction arises because translations are separable from rotations and vibrations, but rotations and vibrations are coupled for general systems. The essential problems Sutcliffe points out undoubtedly exist, and the TRF treatment for nonrigid rotator systems cannot rigorously succeed in general cases, nor can it do so in the NOMO theory. We have focused on the locality of the Gaussian functions, of which the orbital centers can approximately define the rigid-body rotator. It is possible to define center-of-mass (COM), angular, and internal coordinates uniquely for the rigid-body rotator, while it is impossible for the general case where rotational and vibrational motions couple. Thus, the © 2006 American Chemical Society
Published on Web 10/12/2006
First-Order Rovibration Coupling
J. Chem. Theory Comput., Vol. 2, No. 6, 2006 1545
rotational operator has been expanded in a Taylor series with respect to the displacement ∆x based on the rigid-body rotator. As a result, the nuclear wave function represented by Gaussian basis functions can be separated into zeroth-order rigid-body rotation and higher-order coupling. TRF-NOMO theory adopts this unique definition of the COM, angular, and internal coordinates for the zeroth-order rotator.10 In a previous study, however, we implemented a programming code for the TRF-NOMO method corresponding only to the zeroth-order terms of the rotational Hamiltonian and numerically tested their contributions. Therefore, the assessment of the higher-order contribution is of great importance for investigating the reliability of the zeroth-order treatment and, furthermore, the validity of the TRF-NOMO formalism itself, on the basis of the Taylor expansion of the rotational operator. The purpose of the present study is to implement the computational program for the TRF-NOMO method involving the first-order rotational terms, which are much more complicated than the zeroth-order terms, and the numerical assessment of their contribution. The organization of this paper is as follows. First, section 2 describes the theoretical aspects of the first-order TRF-NOMO method. Section 3 indicates the implementation of this method. In section 4, we present the numerical assessments of the present treatment. Concluding remarks are summarized in section 5. Furthermore, the second-order Møller-Plesset (MP2) treatment for the firstorder TRF-NOMO method is described in the Appendix.
2. Theory In this section, we summarize TRF-NOMO/Hartree-Fock (HF) theory,10 which determines electronic and nuclear wave functions simultaneously while eliminating translational and rotational motions. The total Hamiltonian adopted in the original NOMO theory contaminates translational and rotational motions. Thus, the Hamiltonian is called the translation- and rotation-contaminated (TRC) Hamiltonian: H ˆ TRC ) Tˆ e + Tˆ n + Vˆ ee + Vˆ en + Vˆ nn
(1)
and q in eqs 2, 4, and 5 runs over electrons, that of P and Q in eqs 3, 5, and 6 runs over nuclei. We proposed a scheme to eliminate the contribution of translational motion from the TRC Hamiltonian.7 By subtracting the translational Hamiltonian Tˆ T from H ˆ TRC, the TF Hamiltonian is given by H ˆ TF ) H ˆ TRC - Tˆ T where Tˆ T ) -
1
1
∇(xµ)2 - ∑ ∇(xµ) ∇(xν) ∑ 2M µ Mµ
